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# The Mathematician John Forbes Nash Jr.

## Early Life

Born in Bluefield, West Virginia in 1928, John was the son of an electrical engineer who worked for the Appalachian Electric Power Company, while his mother was a former school teacher. He also had a younger sister, Martha, who was born two years after him.

Nash went to public school for most of his life, and he supplemented his formal education through all the books his parents and grandparents gave him. As he got older and his parents began to realize his potential, they helped him further his education by enrolling him in a few classes at their local community college when he was in the final year of high school. Nash studied advanced mathematics at college during that year.

## John Nash 1950 Graduation

## Undergraduate and Graduate School

After completing high school, he received a full scholarship to attend the Carnegie Institute of Technology, with his chosen major being chemical engineering. He quickly switched his major to chemistry after a few months at college, before settling on mathematics after having a few important conversations with his favorite teacher, John Lighton Synge.

Nash had always been a fast learner, which helped him graduate college with both his Bachelor and Master of Science at the age of 19. After his undergraduate studies were complete, Nash attended Princeton University to pursue a graduate degree in mathematics. Even though he was such a young man at the time, all his professors and advisers could see his potential. This is evident from the letter of recommendation written for Nash by his adviser Richard Duffin, who called Nash a “mathematical genius.”

Picking a college for his graduate studies was a tough decision for Nash, because he gained acceptance to both Princeton and Harvard. However, the chair of the mathematics department at Princeton desperately wanted Nash to attend his school, offering him the John S. Kennedy scholarship, which showed Nash that Princeton was more interested in having him as a student. And it was at Princeton where Nash started to work on the advancement of game theory.

Even though the vast majority of Nash’s work on modern day game theory took place at Princeton, the origins of his interest in the subject go back to Carnegie, where he took an international economics course and began to learn about the various choices and consequences present in economic theories. He published a paper at Carnegie called “The Bargaining Problem,” which eventually made its way into the *Econonometrica* publication. He continued to develop this interest at Princeton, where he studied non-cooperative games and the theory behind those games.

When Nash earned his Ph.D. from Princeton in 1950, it was largely on the back of his 28-page dissertation on the topic of non-cooperative games. These games are defined as a game where players make decision independently. In other words, players can cooperate, but the cooperation is self-enforced. His paper contained the framework and theory for what we know as the Nash Equilibrium, which is a specific solution-concept for non-cooperative games. In the Nash Equilibrium, it is stated that within a non-cooperative game involving two or more people, where the players know the equilibrium strategies of everyone else who is competing, no player can gain anything by changing their strategy.

## Good documentary film on John Nash

## The Prisoner’s Dilemma

For those who are not familiar with economics, we can dissect the theory in simple terms. Let us assume we have two players, A and B, who are participating in a non-cooperative game. Both players come up with a strategy and are later told about the strategy of the other person. If both players choose to keep their initial strategy, despite knowing what the other person is going to do, they are in Nash Equilibrium. If they think it is possible to benefit from changing their strategy, after learning about the other person’s strategy, they are not in Nash Equilibrium.

Nash’s contribution to mathematics and economics through the Nash Equilibrium is something few scholars will ever match. When he first came up with the theorem, he had no idea of its wide ranging uses and how it would evolve with modern day economics.

The prisoner’s dilemma, which is used to define many economic situations in game theory, is centered on the premise of two prisoners who are both offered the possibility of a lighter jail sentence if they betray the other person. But neither player has a chance to communicate with the other before they make their decision.

Since the prosecutor’s in the example do not have enough evidence to convict both prisoners of the crime, they encourage each of them to betray the other. There are a few different scenarios in the game, depending on the decisions made by both players. If both players betray each other, each of them is going to serve five years in jail. If player A betrays B, but player B remains silent, player A does no time and player B does ten years, and vice versa. If they both remain silent, they will each serve a single year in jail.

In this example, the Nash equilibrium only exists at one point, when they both decide to betray each other. Why is this the Nash equilibrium, even though the scenario where both players remain silent is the optimal choice? Because it is the only point where neither player can gain anything from changing their decision, after learning about the first player’s choice.

Even though the scenario where they both stay silent is optimal in the game, if either player decided to betray the other, based on learning the other person was silent, that player would serve no jail time. The promise of lesser jail time would prompt either player to change their mind after learning of the other’s choice, which is why this scenario is not a Nash equilibrium.

The example of the Nash Equilibrium as illustrated in the prisoner’s dilemma can be applied to various real-world economic problems.

## Real Algebraic Geometry

Aside from the Nash Equilibrium, John Forbes Nash also made huge contributions to mathematics in the field of real algebraic geometry. He came up with the Nash embedding theorem, which highlights the fact that each abstract Riemannian manifold is isometrically realized as a submanifold of the Euclidean space. His work on the singularity theory also gained him a great deal of acclaim from fellow mathematicians.

There is an interesting story in Nash’s biography, *A Beautiful Mind*, which talks about his process for working on the proof of Hilbert’s nineteenth problem. When working on this theorem, which deals with elliptic partial differential equations, Nash discovered that Ennio de Giorgi, a fellow mathematician, had already published a proof of the same equation a few months before Nash could finish his own proof. Even though Nash had used a different method for coming to the same conclusion, he was understandably disappointed. The two men made each other’s acquaintance in New York in 1956 and the encounter was said to be fairly amicable. Mathematicians at the time speculated that if either Giorgi or Nash had never completed the proof, the other man would have won a Fields Medal for their work.

Nash also did some great work with regards to encryption, with some letters from the National Security Agency showing that they had corresponded with the mathematician many times during the 1950s. Nash had proposed a different type of encryption in the letters – a decryption machine. Even though he was not directly involved in the creation of any decryption methods, his letters also showed that he was already thinking about many concepts regarding cryptography that had not even been invented yet.

## Website explaining some of Nash's mathematics

## Biography of John Nash Jr.

## Fight with Mental Illness

After leaving Princeton, Nash took up a position at the mathematics faculty at the Massachusetts Institute of Technology. He also met his first serious girlfriend, a nurse named Eleanor Stier, when he was admitted to a local clinic as a patient. However, Nash chose not to remain with Stier when he discovered that she was pregnant, because he was angered by the pregnancy and the fact that her social status did not match his, which made them a poor match in his eyes.

One of the unfortunate facts about the life of John Forbes Nash Jr. is that he struggled with a mental illness for many years. The first signs of his illness came in the form of intense paranoia, with Nash convinced that any man who wore a red tie was part of a communist agenda against him! He even sent letters to various embassies and government offices in Washington D.C., telling them that these men in red ties were trying to make their own government.

Nash’s mental illness did not start becoming apparent until 1958. His erratic behavior was becoming apparent to his wife, Alicia Lopez-Harrison de Lardé, and his co-workers at MIT. The problem would rear its head when he delivered a lecture at Columbia University in 1959. Nash was due to speak about the Riemann hypothesis proof that he had completed, but his lecture was completely incoherent. People who were watching him speak could clearly see that something was wrong with this brilliant man. After the incident, Nash was admitted to a hospital for tests and checks to see what was going on in his head. He was eventually diagnosed as a paranoid schizophrenic, which explained his intense paranoia and periodically erratic behavior.

After his hospitalization, he resigned from MIT and stayed home with Alicia. The seclusion of home life helped Nash understand and deal with some of his psychotic delusions and episodes he was experiencing. During two particularly intense schizophrenic episodes, Nash traveled to Europe. Driven by paranoia and voices in his head he sought to renounce his US citizenship and become a citizen of the world. With the help of government officials, John was safely returned to the United States after both episodes.

During the 1961 to 1970 John would be hospitalized serval times for his schizophrenia. During one stay at the Trenton State Hospital, he was subjected to insulin therapy, which was both very dangerous and provided little relief for his condition. Most of the treatments Nash received during those years did not have much of an impact on his mental state, which prompted Nash to vow in 1970 that he would never return to a hospital.

Then, in the early 1980s, John mental illness began to improve. The change was slow, but over time his lucid thoughts returned. Nash would later write of his condition, “Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically-oriented thinking as essentially a hopeless waste of intellectual effort.”

Nash’s later writings confirmed that he would only sporadically take his prescribed medications. He believed that the benefits of psychotropic drugs were vastly overrated, while he also felt as though the side-effects of these drugs were too much for him to handle.

## Awards and Recognition

John Nash won many awards and accolades over the years. He received the John von Neumann Theory Prize in 1978 for discovering non-cooperative equilibria, which we call Nash equilibria. The big prize came in 1994 when John F. Nash Jr. received the SverigesRiksbank Prize in Economic Sciences in Memory of Alfred Nobel, alongside German professor Reinhard Selten and John C. Harsanyi from the University of California, Berkeley. This prestigious award was presented to the three men for their research on game theory. Although certain groups questioned the Nobel Committee for granting the award to a person who suffered from severe mental illness, Selten and Harsanyi defended Nash. The three winners split a prize worth $930,000.00. The prize citation recognized the “pioneering analysis” of all three Nobel Laureates in the field of game theory. Assar Lindbeck, the former chair of the committee for the Nobel Prize in Economic Sciences, remarked about Nash, “We lifted him into daylight...We resurrected him in a way.”

The biography of Nash’s life, *A Beautiful Mind*, written by Sylvia Nasar was the basis of a movie by the same name. The movie starred Russell Crowe and won four Oscars, including the Academy Award for Best Picture.

## "A Beautiful Mind" Movie Trailer

## John Nash at a conference in 2011

## Death

John Forbes Nash Jr and his wife Alicia were both killed in a car ancient on May 23, 2015 on their return trip from Norway. The couple had traveled to Norway so John could receive the Abel Prized from the King of Norway. They had just landed at the Newark airport and were traveling home when their taxi driver struck a guardrail, with the accident resulting in both Nash and his wife being ejected from the car.

For all those who have found inspiration in John Nash’s struggles with mental illness, the mathematician serves as a beacon of hope. His difficult but triumphant recovery is a testament to the power of resolve in overcoming paranoid schizophrenia, a debilitating mental illness.

## Further Reading

*John Nash, Jr. – The Life and Legacy of One of America’s Most Influential Mathematicians*. Charles Rivers Editors. 2015.

Nasar, Sylvia. *A Beautiful Mind*. Simon & Schuster. 2011.

John Nash (Author), Harold William Kuhn (Editor), Sylvia Nasar. *The Essential John Nash*. Princeton University Press. 2001.